
Therefore, our theory recovers the inter-layer term in the BM continuum model [1]}, [2]}, [3]}
if we set \(w_3=0\)  and keep only \(q\) s in the first shell and neglect the gradient couplings \(\Lambda _{SS^{\prime }}\) . Furthermore, in the absence of the lattice relaxation, as derived in the appendix [4]}, our theory gives \(w_0 = w_1\)  [1]}. As shown in the next section, the \(w_3\)  term is responsible for non-negligible p-h asymmetry for the model of Ref. [6]}.
